Lesson 11 AC Circuits AC Ciruits Power Maximum and Instantaneous voltage drops and current ...
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Transcript of Lesson 11 AC Circuits AC Ciruits Power Maximum and Instantaneous voltage drops and current ...
![Page 1: Lesson 11 AC Circuits AC Ciruits Power Maximum and Instantaneous voltage drops and current Phasor Diagrams Phase angle RLC Circuits Resonance.](https://reader035.fdocuments.us/reader035/viewer/2022062222/5697bfcf1a28abf838ca9d54/html5/thumbnails/1.jpg)
Lesson 11Lesson 11Lesson 11
AC Circuits
AC CiruitsPowerMaximum and Instantaneous voltage drops and currentPhasor DiagramsPhase angle
RLC Circuits Resonance frequency
High and Low pass filtersStep up and Step down Transformers
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AC GeneratorAC Generator
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AC emf source
t N ddt
N ddt
B A
Nd
dtBACos NBA d
dtsin t
if rotational speed is constant: t t t NBA sin t
maxsin t
max
NBA
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rms current
Effective (Integrated ) values of I and V
i t Imax sin t
2 f 2 T
; T is the period of oscillation
Instantaneous power i t v t
Heat dissipated = Power used in load i t 2 R Im2Rsin
2 t Average power over one cycle
Pave 1
TIm
2Rsin
2 t 0
T
dt Im2R
1
Tsin
2 t 0
T
dt Im
2R
2
Define
Pave Irms2R Irms
Im2
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Alternating Current Circuits
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t)
Veff and v(t)
Ieff and i(t)
ac-R circuit
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Phasor diagram for R iR(t)
t
iR(t)IRmsin(t)=
Phasor Diagram
Current through Load Resistance
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Phasor diagram for R cont.vR(t)
t
vR(t)VRmsin(t)=
Phasor Diagram
PD across Load Resistance
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iR t 1
RvR t
t R
Instantaneous current and voltage
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ac-C circuit
t)
vC(t)
iC(t)
vC(t)
t
iC(t)
Current in Circuit and PD across
Capacitor
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ac-L circuit
vL(t)
t
iL(t)
t)
vL(t)
iL(t)
Current in Circuit and PD across
Inductor
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The phase angle between the current and the voltage:
In the resistor is 0 radIn the capacitor is - rad
( Current Ahead)In the inductor is + rad
(Current Behind)
Summary
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Series RLC circuit Series RLC circuit series ac-RLC circuit
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Instantaneous currentCurrent through all elements is the same
i t iV t iC t iL t Thus the instantaneous PD’s
must be out of phase
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Picture
Total Potential
Drop across R, L & C.
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4 4
1
L CR
L C
Rm Lm Cmm
L C
L
C
T Tv t v t
v ti t
R X X
V V VI
R X X
X L
XC
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Phasor Diagram for RLC circuit IvL(t)
vR(t)
vC(t)
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Instantaneous Potential Change Across R, L & C
v t vR t vL t vC t t Thus circuit phasor is
v RLC t v R(t) v L (t) v C(t)
Instantaneous PD
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Phasor Diagram for RLC circuit II
vL(t)
vR(t)vC(t)
vRLC(t)
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Instantaneous PD as projection onto y-axis
vRLC(t)
v(t1)v(t2)
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Phase Angle
Phase Angle
vRLC(t)
tan VLm VCmVRm
Im XL Im XCIm R
XL XCR
tan1 XL XC
R
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series ac-RLC graph
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ImpedanceThe magnitude of the Total Potential Phasor is
Vm VR2 VLm VCm 2
Im2R
2 ImXL ImXC 2
Im R2 XL XC 2
ImZ
Impedance : Z R2 XL XC 2
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Table of definitions
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Impedance and reactance
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Im VmZVRm
RV Lm
XLVCmXC
Generalized Ohm's Law.
Impedance Z R2 XL XC 2
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1tan L CX X
R
Phase Angle between total PD across circuit
and the current
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Power Factor
Power is only used in AC circuit in load resistance
P t i t 2
R
(energy is not used in inductor or capacitor)
I m2sin
2 t - R(current is always in phase with PD across total resistance)
P ave = I rms2R I m
2
2R
rms
Z
I rms R rms I rms
RZ rms I rms cos
cos Power Factor
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Angular frequency dependence
Power and current depend on angular
frequency of circuit
Power and current depend on angular
frequency of circuit
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Max I ; Min ZIm Vm
Z
mZ
Z R2 XL XC 2
R2 L 1
C
2
R2
2LC 1C
2
Z is a minimum when 2 LC 1 0
which occurs when
0 1LC
XL XC
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Pave 1
2Im
2 R 1
2
Vm2R
Z 2
1
2
Vm2R
R2 L 1
C
2 1
2
Vm2R
R2 L
2
2 2 0
2 2
1
2
Vm2R
2
R2 2 L
2 2 0
2 2
Power as a function of
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ResonanceCircuit uses most power / current
when it is in RESONANCE with applied frequency
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Imax and Pave versus
Im Pave
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Width of Power curve is a measure of the QUALITY
of the circuitSmall width - High QualitySharpness of response to
external frequency
Quality of circuit
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RC Filters IRC Filters
VVoutout
VVinin
Low Pass Filter
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VoutVin
ImXCImZ
XCZ
1C
R2 1
C
2
1
RC 2 2 1 then Vout Vin
Vout (as Vin const.)
Low pass Filter
Low Pass Filter
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RC Filters
VVoutout
VVinin
RC Filters IIHigh Pass Filter
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VoutVin
ImRImZ
RZ R
R2 1
C
2
then Vout Vin Vout (as Vin const.)
High pass Filter
High Pass Filter
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Transformers IStep up and Step
down Transformers
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Transformers IIV 1 N 1
d Bdt
V 2 N 2
d Bdt
Fluxes are the same
V 2 N 2
N 1
V 1